NEMATIC ELASTOMERS
VIRGINIA AGOSTINIANI, ANTONIO DESIMONE, AND KONSTANTINOS KOUMATOS
Abstract. Using the theory of Γ-convergence, we derive from three-dimensional elasticity new one-dimensional models for non-Euclidean elastic ribbons, i.e. ribbons exhibiting spontaneous curvature and twist. We apply the models to shape-selection problems for thin films of nematic elastomers with twist and splay-bend texture of the nematic director. For the former, we discuss the possibility of helicoid-like shapes as an alternative to spiral ribbons.
MSC (2010): 74B20, 76A15, 74K10, 74K20.
Keywords: nematic elastomers, dimension reduction, rod theory.
1. Introduction
Shape morphing systems are common in Biology. They are used to control locomotion in unicellular organisms [8, 9] and to produce controlled motions in plants [6, 13, 18, 21, 33]. Differential swelling and shrinkage processes, partially hindered by fibers, lead to dynamical conformation changes which are essential in the life of many botanical systems [30]. Inspired by Nature, many attempts have been reported in the recent literature to engineer artificial shape-morphing systems based on synthetic soft materials [22, 26, 31] and the interest in the general topic of “shape programming” is steadily growing.
A useful tool has emerged in the mathematical literature to describe the me-chanics of shape programming, namely, non-Euclidean structures (non-Euclidean plates and rods). These are elastic structures described by functionals which are minimised by configurations exhibiting nonzero curvature. The relevant energy functionals are often postulated on the basis of physical intuition [24] but, in more recent attempts, they are derived from three-dimensional models [4, 25, 29] through rigorous dimension reduction based on the theory of Γ-convergence, following the approach pioneered in [1, 20].
Liquid crystal elastomers (LCEs) provide an interesting model system for the study of shape programming. They are polymeric materials that respond to exter-nal stimuli (temperature, light, electric fields) by changing shape [2, 3, 5, 14, 15, 35] and are typically manufactured as thin films [7, 10, 11, 12]. Suitable textures of the nematic director imprinted at fabrication lead to thin structures with tunable and controllable spontaneous curvature, see [4, 27, 28, 34]. In particular, for the twist geometry (nematic director always parallel to the mid-plane of the film and rotating by π/2 from the bottom to the top surface of the film), it has been ob-served both experimentally and computationally [28, 32] that, depending on the aspect ratio of the mid-plane, either spiral ribbons (this is the case of large width over length aspect ratio) or helicoid-like shapes (this is the case of small width over length aspect ratio) emerge spontaneously.
In this paper, we provide a rigorous mathematical description of thin structures made of nematic elastomers (both in the case of twist and of splay-bend geometry) where the minimisers of the deduced energy functionals reproduce the experimen-tally observed configurations. Our analysis stems from the combination of two
main results: first, we use the 3D-to-2D dimension reduction result in [4] for (non-Euclidean) thin films of nematic elastomers, and, secondly, we use a non-Euclidean version of the 2D-to-1D dimension reduction result of [19], where a corrected version of the well-known Sadowsky functional is derived for the mechanical description of inextensible elastic ribbons. The reader is also referred to [17, 23] and all other papers in the same special issue of the Journal of Elasticity for more material on the mechanics of elastic ribbons.
Our results show that the technique of rigorous dimensional reduction based on Γ-convergence, far from being just a mathematical exercise, can provide a tool to derive, rather than postulate, the functional form and the material parameters (elastic constants, spontaneous curvature and twist, etc.) for dimensionally reduced models of thin structures. For example, the possibility of assigning an energy to helicoid-like shapes - which in the 1D reduced model are represented by rods with straight mid-line, zero flexural strain around the width axis (in short, flexure) and nonzero torsional strain (in short, torsion), see the second picture in Figure 4 -rests precisely on the fact that, in the narrow ribbon limit, the isometry constraint on the mid-plane of the 2D theory is lost. This is the origin of the “correction” [19] to Sadowsky’s functional (visible in the regime where flexure α is smaller than torsion β in formula (3.8)), a correction that can only be obtained with a variational notion of convergence of energy functionals (Γ-convergence). We concentrate our discussion on liquid crystal elastomers, but clearly our method is applicable to more general systems, whenever differential spontaneous distortions in the cross section induce spontaneous flexure and torsion of the mid-line of the rod (see e.g. [31]).
The starting point of the subsequent analysis is a family of non-Euclidean plate models defined on a narrow strip of width ε cut out from the plane (e1, e2) at an
angle θ with the horizontal axis (see energy (2.5) below). In Section 2 we set-up our 2D model and show that minor modifications of the results of [19] allow us to derive, in the limit as ε ↓ 0, the 1D model defined in (2.12)–(2.13). Again in Section 2, we observe that examples of our starting 2D theory are given by twist and splay-bend nematic LCE sheets, as obtained from 3D nonlinear elasticity in [4]. The special role of the e1 basis vector emerges from these concrete examples
as the direction of the nematic director in the bottom face of the LCE sheet, see Figure 1. The limiting 1D theory, which is a non-Euclidean rod theory, is then explicitly computed for twist and splay-bend LCE ribbons in Sections 3 and 4, respectively. In particular, the explicit expression of the limiting energy densities of the rods, depending on the flexural strain around the width axis and the torsional strain, and their minimisers, are given in Proposition 3.2 and Lemma 3.3 (in the twist case) and in Proposition 4.1 and Lemma 4.2 (in the splay-bend case). These minimisers represent spontaneous flexure and torsion of LCE rods and their explicit formulas are provided in some special cases in Remarks 3.1 and 4.2. Starting from these computed values of spontaneous flexure and torsion, we reconstruct the non-trivial configurations assumed by these rods in the absence of external loadings, see Figures 3 and 4, which have been observed in experiments [27, 28, 34].
2. A non-Euclidean Sadowsky functional
Let ω be an open planar domain of R2. In the framework of a nonlinear plate
theory [20], we consider the bending energy ˆ ω n c1|Avˆ(ˆz) − ¯A|2+ c2tr2(Avˆ(ˆz) − ¯A) + ¯e o dˆz (2.1)
associated with a developable surface ˆv(ω), ˆv being a deformation from ω to R3. In the previous expression, Aˆv(·) ∈ R2×2sym denotes the second fundamental form of
ˆ
v(ω), c1 > 0 and c2 > 0 are material constants, and ¯A ∈ R2×2sym and ¯e represent
a characteristic target curvature tensor and a characteristic nonnegative energy constant, respectively. Although the constant ¯e is inessential in the problem of finding minimal energy configurations, we prefer not to drop it in order to emphasize that, due to the kinematic incompatibility of the spontaneous strains in the 3D parent model, elastic energy due to residual stresses will always be present in our system. In other words, all the non-Euclidean plate and rod models in this paper describe systems which are never stress-free. Moreover, the notation tr2A stands for the square of the trace of A. We recall that Avˆ can be expressed as (∇ˆv)T∇ˆν,
where ˆν = ∂z1ˆv ∧ ∂z2v. In [4], the two-dimensional energy (2.1) has been rigorouslyˆ
derived from a three-dimensional model for thin films of nematic elastomers with splay-bend and twist orientation of the nematic directors along the thickness and the following explicit formulas have been obtained for ¯A
¯ AS = k diag(−1, 0), A¯T = k diag(−1, 1), k := 6 η0 π2h 0 , (2.2) and for ¯e ¯ eS = µ (1 + λ) π4− 12 32 η2 0 h2 0 ¯ eT = µ π4− 4π2− 48 8π4 η2 0 h2 0 . (2.3)
We recall that, in the splay-bend and in the twist geometry, the nematic director continuously rotates by π/2 from being parallel to e1 at the bottom face to being
parallel to e3 and e2, respectively, at the top face, see the first two pictures of
Figure 1. Here and throughout the paper we use the indices “ S ” and “ T ” for the quantities related to the splay-bend and the twist case, respectively. In the previous formulas, η0 is a positive dimensionless parameter quantifying the magnitude of
the spontaneous strain variation along the (small) thickness h0 of the film, µ is the
elastic shear modulus, and λ + 2µ/3 is the bulk modulus. We remark that nematic LCEs are anisotropic materials: both the energy well structure and the elastic moduli are, in principle, anisotropic. Since, however, experimental data on the elastic moduli are not currently available, most 3D models neglect the anisotropy of the elastic constants since many of the observed effects of anisotropy are already accounted for by the (anisotropic) structure of the energy wells (see e.g. [16]). This leads to anisotropic 2D energies such as (2.1), in which the anisotropy is confined to the target curvature, while the elastic constants only contain the Lam´e coefficients typical of an isotropic material, see [4]. Also, the material constants appearing in (2.1) are given by c1= µ 12, c2= λµ 12. (2.4)
We refer the reader to [4] for a detailed description of the three-dimensional model and of the splay-bend and twist nematic director fields. In Sections 3 and 4 we specialize our results to the case where the curvature tensor ¯A is of the form (2.2), while in the rest of this section we focus on a general energy density of type (2.1). Denoting by {e1, e2} the canonical basis of R2, we cut out of the planar region
ω a narrow strip Sεθ:=nz1eθ1+ z2eθ2: z1∈ (−`/2, `/2), z2∈ (−ε/2, ε/2) o ⊂ ω, 0 ≤ θ < π, with eθi := Rθei, i = 1, 2, Rθ= cos θ − sin θ sin θ cos θ ,
e e 3 1 e e 1 2 l e ε θ e 1 2
Figure 1. Top two pictures: distribution of the nematic directors along the thickness of the sheet, in the splay-bend and in the twist geometry, respectively. Bottom: a narrow strip Sθ
ε cut out of the
mid-plane of the sheet, at an angle θ with the e1-direction.
see the third picture of Figure 1, and consider the energy (2.1) restricted to the strip Sθ ε, namely ˆ Eθ ε(ˆv) := ˆ Sθ ε n c1|Aˆv(ˆz) − ¯A |2+ c2tr2(Aˆv(ˆz) − ¯A ) + ¯e o dˆz, (2.5) where ˆv : Sθ
ε → R3 is a deformation such that ˆv(Sεθ) is a developable surface. We
are interested in examining the behaviour of the minimisers of the functionals ˆEεθin
the limit of vanishing width, i.e. ε ↓ 0. Notice that using the function v : Sε→ R3
defined in the unrotated strip Sε := S0ε as v(z) = ˆv(Rθz), we have that v(Sε) is
developable and ˆEθ ε(ˆv) can be rewritten as ˆ Eθ ε(ˆv) = ˆ Sε n c1|Avˆ(Rθz) − ¯A |2+ c2tr2(Aˆv(Rθz) − ¯A ) + ¯e o dz = ˆ Sε n c1|Av(z) − RTθAR¯ θ|2+ c2tr2(Av(z) − ¯A ) + ¯e o dz,
where in the second equality we have used the fact that Aˆv(Rθz) = RθAv(z)RTθ.
Now, introducing a suitable rescaling and setting Eθ ε(v) := 1 ε ˆ Sε n c1| Av(z) − ¯Aθ|2+ c2tr2(Av(z) − ¯A ) + ¯e o dz, (2.6) with ¯ Aθ := RTθAR¯ θ, (2.7)
we have that ˆEεθ(ˆv) = εEεθ(v). Having this identification in mind, from now on we always deal with the functional v 7→Eεθ(v). Expanding the integrand we obtain the
following general form of the bending energy Eθ ε(v) = 1 ε ˆ Sε n c|Av(z)|2+ Lθ(Av(z)) o dz, (2.8)
where Lθ is the affine function defined as
Lθ(A) := −2c1A · ¯Aθ− 2c2trA tr ¯Aθ+ c1| ¯Aθ|2+ c2tr2A¯θ+ ¯e. (2.9)
In the above expression, we have set c = c1+ c2 and we have made use of the fact
that
tr2A = |A|2+ 2 det A, for all A ∈ R2×2
sym. In fact, since v is an isometry of the planar strip Sε, the Gaussian
curvature associated with v(Sε) vanishes, i.e.
det Av(z) = 0.
The natural function space for v is the space of W2,2 isometries of Sε defined as
W2,2iso(Sε, R3) :=
n
v ∈ W2,2(Sε, R3) : ∂iv · ∂jv = δij
o . In order to express the energy over the fixed domain
S = I ×−1 2, 1 2 , I :=−` 2, ` 2 ,
we change variables and define the rescaled version y : S → R3of v, given by
y(x1, x2) = v(x1, εx2).
The following procedure is rather standard and we use the notation of [19] as, in the sequel, our proofs will be largely based on this paper. By introducing the scaled gradient
∇ε· = (∂1· |ε−1∂2·)
we obtain that ∇εy(x1, x2) = ∇v(x1, εx2) and y belongs to the space of scaled
isometries of S defined as
W2,2iso,ε(S, R3) := ny ∈ W2,2(S, R3) : |∂1y| = |ε−1∂2y| = 1, ∂1y·∂2y = 0 a.e. in S
o . Similarly, we may define the scaled unit normal to y(S) by
ny,ε= ∂1y ∧ ε−1∂2y
and the scaled second fundamental form associated to y(S) by Ay,ε= ny,ε· ∂1∂1y ε−1ny,ε· ∂1∂2y ε−1ny,ε· ∂1∂2y ε−2ny,ε· ∂2∂2y .
With this definition, Ay,ε(x1, x2) = Av(x1, εx2) and Eεθ(v) = Jεθ(y), where the
functional Jθ ε(y) := ˆ S n
c|Ay,ε|2+ Lθ(Ay,ε)
o
dx (2.10)
is defined over the space W2,2iso,ε(S, R3) of scaled isometries of S.
Lemma 2.1 (Compactness). Suppose (yε) ⊂ W2,2iso,ε(S, R3) satisfy
sup
ε J
θ
ε(yε) < ∞.
Then, up to a subsequence and additive constants, there exist a deformation y ∈ W2,2(I, R3) and an orthonormal frame (d1|d2|d3) ∈ W1,2(I, SO(3)) fulfilling
d1= y0 and d01· d2= 0 a.e. in I,
and such that
yε* y in W2,2(S, R3), ∇εyε* (d1|d2) in W1,2(S, R3×2).
Moreover, for some γ ∈ L2
(S, R3), Ay,ε* d01· d3 d02· d3 d02· d3 γ in L2(S, R2×2sym).
Proof. Note that Jθ ε(yε) = ˆ S n c1|Ayε,ε− ¯A θ|2+ c 2tr2 Ayε,ε(x) − ¯A θ + ¯eodx ≥ ˆ S c1|Ayε,ε(x) − ¯A θ|2dx.
But, since ¯Aθ is constant, this implies that kAyε,εk 2
L2(S,R3)is bounded uniformly in
ε and the proof is identical to the proof of Lemma 2.1 in [19]. In order to state the Γ-convergence result, we define
A := n(d1, d2, d3) : (d1|d2|d3) ∈ W1,2(I, SO(3)), d01· d2= 0 a.e. in I
o
(2.11) and the functionalJθ
: A → R by Jθ(d 1, d2, d3) := ˆ I Qθ(d01· d3, d02· d3) dx1, (2.12) where Qθ(α, β) := min γ∈R c|M |2+ 2c| det M | + Lθ(M ) : M = α β β γ , (2.13) and Lθ(M ) is defined according to (2.9). We recall that the constraint d01· d2= 0
means that the narrow strip does not bend within its plane or, equivalently, that there is no flexure around the direction of d3. Moreover, the quantities d01· d3and
d02·d3have the physical meaning of flexural strain around d2and of torsional strain.
To be short, throughout the paper we refer to them as flexure and torsion. The proposition below provides an alternative expression for Qθ which is used heavily in the sequel. For the ease of the reader, we postpone its proof until the end of this section.
Proposition 2.2. Qθ is a continuous function given by
Qθ(α, β) = −l21 4c− l1α + l2, in D 4cβ2−l21 4c+ l1α + l2, in U cα2+βα 2 2 + l1β 2 α + l2, in V, (2.14)
where l1= l1( ¯Aθ), l2= l2(α, β, ¯Aθ) is an affine function of α and β, and
D := (α, β) ∈ R2: −l1 2cα > β 2+ α2 , (2.15) U := (α, β) ∈ R2: −l1 2cα ≤ β 2− α2 , (2.16) V := R2\ (D ∪ U ). (2.17)
Theorem 2.3 (Γ-convergence). The functionals Jθ
ε Γ-converge to Jθ as ε → 0
in the following sense:
(1) (Γ-lim inf inequality) for every sequence (yε) ⊂ W 2,2
iso,ε(S, R
3), y ∈ W2,2
(I, R3)
and (d1, d2, d3) ∈ A with y0 = d1 a.e. in I, yε * y in W2,2(S, R3) and
∇εyε* (d1|d2) in W1,2(S, R3×2), we have
lim inf
ε→0 J
θ
(2) (recovery sequence) for every (d1, d2, d3) ∈ A there exists (yε) ⊂ W 2,2
iso,ε(S, R
3)
and (up to an additive constant) y satisfying y0 = d1 such that yε * y in
W2,2 (S, R3), ∇ εyε* (d1|d2) in W1,2(S, R3×2), and lim ε→0J θ ε(yε) =Jθ(d1, d2, d3).
In proving the existence of a recovery sequence, we will need the following lemma which is a slight variation of [19, Lemma 3.1].
Lemma 2.4. For every M ∈ L2
(I, R2×2
sym) there exists a sequence (Mn) ⊂ L2(I, R2×2sym)
satisfying det Mn= 0 a.e. in I and for all n ∈ N such that Mn * M in L2(I, R2×2sym)
and ˆ I c|Mn|2+ Lθ(Mn) dx1−→ ˆ I c|M |2+ 2c| det M | + Lθ(M ) dx 1.
Proof. The construction of the sequence Mnis identical to that in [19, Lemma 3.1].
To pass to the limit as n → ∞, simply note that ˆ I Lθ(Mn)dx1−→ ˆ I Lθ(M )dx1, since Lθ is affine. Proof of Theorem 2.3.
(1) (Γ-lim inf inequality) Let (yε) ⊂ W2,2iso,ε(S, R3), y ∈ W2,2(I, R3) and (d1, d2, d3) ∈
A with y0= d
1a.e. in I, yε* y in W2,2(S, R3) and ∇εyε* (d1|d2) in W1,2(S, R3).
We may assume that lim infεJεθ(yε) < ∞, as otherwise the result follows trivially,
and by passing to a subsequence that supεJθ
ε(yε) < ∞.
Lemma 2.1 now states that Aε:= A
yε,ε* A in L 2 (S, R2×2 sym) where A = d01· d3 d02· d3 d0 2· d3 γ . We may now estimate
lim inf ε J θ ε(yε) = lim inf ε ˆ S c|Aε|2dx + lim ε ˆ S Lθ(Aε) dx ≥ ˆ S
c|A|2+ 2c| det A| +
ˆ
S
Lθ(A) dx ≥Jθ(d
1, d2, d3),
where the first integral is estimated exactly as in [19] and, since Lθ is affine, the
second integral is weakly continuous.
(2) (recovery sequence) Fix (d1, d2, d3) ∈ A and y ∈ W2,2(I, R3) such that y0 = d1
a.e. in I. Define R := (y0|d2|d3) ∈ SO(3) a.e. in I and
M = y00· d3 d02· d3 d02· d3 γ ,
where, for a.e. x1 ∈ I, γ(x1) is chosen to be the value realising the minimum in
(2.13) with α = y00(x1) · d3(x1), β = d02(x1) · d3(x1). Namely,
Qθ(y00· d3, d20 · d3) = c|M |2+ 2c| det M | + Lθ(M ) a.e. in I.
Note that, due to the precise form of γ given by (2.18) in the proof of Proposition 2.2 at the end of this section, we have that
γ = (−l1/2c − M11)χ −l12cM11>M122+M112 + (−l1/2c + M11)χ −l12cM11≤M122−M112 + M 2 12 M11 χ M2 12−M112<−l12cM11≤M122+M112
where M11 := y00· d3, M12 := d02· d3, and l1 is a θ-dependent constant. From this
fact, it is easy to deduce that γ ∈ L2(I), considering that M
11∈ L2(I).
Through Lemma 2.4 we may then find a sequence (Mn) ⊂ L2(I, R2×2sym) such that
det Mn= 0, Mn * M in L2(I, R2×2sym) and, as n → ∞,
ˆ I c|Mn|2+ Lθ(Mn) dx1−→ ˆ I c|M |2+ 2c| det M | + Lθ(M ) dx 1.
Performing the same construction as in [19], we find a sequence yj
εwith the property
that the associated second fundamental forms Ayj
ε,ε satisfy
Ayj ε,ε → M
j strongly in L2
(S, R2×2sym) as ε → 0,
where the matrices Mj have been obtained by truncating and mollifying Mn and
themselves satisfy Mj * M in L2(I, R2×2sym) as well as
ˆ I c|Mj|2dx1−→ ˆ I c|M |2+ 2c| det M | dx 1.
Then, since Lθ is affine, we deduce that
lim ε→0J θ ε(y j ε) = lim ε→0 ˆ S h c|Ayj ε,ε| 2+ Lθ(A yjε,ε) i dx = ˆ S c|Mj |2+ Lθ(Mj) dx j→∞ −→ ˆ S c|M |2+ 2c| det M | + Lθ(M ) dx = ˆ S Qθ(d01· d3, d02· d3) dx1=Jθ(d1, d2, d3).
The proof can then be concluded by taking diagonal sequences. We end this section with the proof of Proposition 2.2.
Proof of Proposition 2.2. For a matrix M = α β β γ , the expression for Qθ in (2.13) becomes
Qθ(α, β) = min
γ∈Rf (γ),
where
f (γ) := c(α2+ 2β2+ γ2) + 2c|αγ − β2| + l(γ). Here, l is an affine function of γ
l(γ) = l1γ + l2,
with l1 = l1( ¯Aθ), and l2 = l2(α, β, ¯Aθ) affine in α and β. Note that if α = 0, f
reduces to the differentiable function
f (γ) = 4cβ2+ cγ2+ l(γ) and it is minimised at γ = −l1
2c, i.e. for all β ∈ R
Qθ(0, β) = 4cβ2− l
2 1
4c+ l2. Next assume that α 6= 0. If γ = β2/α,
f (β2/α) = cα 2+ β2 α 2 + l1 β2 α + l2
and for any γ 6= β2/α, the function f is differentiable with f0(γ) = 2cγ + 2cα sgn(αγ − β2) + l1, which vanishes if γ = −l1 2c− α sgn(αγ − β 2). If αγ > β2, then γ
1= −l2c1 − α is the critical point and this can only be true in the
regime
−l1 2cα > β
2+ α2.
In this case, we compute
f (γ1) = −
l2 1
4c− l1α + l2.
Similarly, for αγ < β2, we find that γ2 = −2cl1 + α is the critical point which can
only be true in the regime
−l1 2cα < β 2− α2 and then f (γ2) = 4cβ2− l2 1 4c+ l1α + l2. On the other hand, in the regime
β2− α2≤ −l1
2cα ≤ β
2+ α2,
a straightforward computation shows that f0(γ) < 0 if γ < β2/α and f0(γ) > 0 if γ > β2/α. Hence, in this regime, and with α 6= 0, the minimum value of f is achieved at γ = β2/α so that Qθ(α, β) = f (β2/α) = cα 2+ β2 α 2 + l1 β2 α + l2.
To compute Qθ in the respective regimes of values of l1α, one needs to understand
whether the value of f at its respective local minima γ1 and γ2 is lower than
f (β2/α). We compute f (γ1) − f (β2/α) = −c ( l2 1 4c2 + α2+ β2 α 2 + l1 α2+ β2 α ) = −cα 2+ β2 α + l1 2c 2 ≤ 0 with equality if and only if
−l1 2cα = β 2+ α2. Similarly, f (γ2) − f (β2/α) = −c ( l21 4c2 + α2− β2 α 2 − l1 α2− β2 α ) = −cα 2− β2 α − l1 2c 2 ≤ 0 with equality if and only if
−l1 2cα = β
2
Hence, we deduce the result. Note that these computations show that Qθis contin-uous and that the value of γ = γ(α, β, θ) realising the minimum in (2.13) is given by γ(α, β, θ) = −l1/2c − α, in D −l1/2c + α, in U β2/α, in V, (2.18) where the sets U , D, and V are defined in (2.16)–(2.17).
3. The twist case
In the case where the (physical) energy (2.5) is derived from a three-dimensional model with a twist-type nematic director field imprinted in the thickness of a LCE thin film, the characteristic quantities ¯A and ¯e present in (2.5) are given by ¯AT and
¯
eT in formulas (2.2) and (2.3). Correspondingly, the θ-dependent target curvature
tensor ¯Aθdefined in (2.7) becomes ¯ AθT = k −aθ bθ bθ aθ , aθ:= cos 2θ, bθ:= sin 2θ, θ ∈ [0, π), (3.1) and tr ¯Aθ
T = tr ¯AT = 0. Moreover, the functionalEεθ defined in (2.8) in this case
reads Eθ ε,T(v) := 1 ε ˆ Sε n c|Av(z)|2+ LθT(Av(z)) o dz, with LθT(A) := −2c1A · ¯ATθ + 2c1k2+ ¯eT.
Also, the energy Jθ
ε defined in the rescaled configuration S (see (2.10)) is
Jθ ε,T(y) :=
ˆ
S
n
c|Ay,ε(x)|2+ LθT(Ay,ε(x))
o
dx, (3.2)
for every y ∈ W2,2iso,ε(S, R3). We recall that ˆEεθ(ˆv) = εEεθ(v) = εJεθ(y), where ˆ
v : Sθ
ε → R3, v : Sε→ R3, y : S → R3 are isometries and are related to each other
via the following relations
v(z) = ˆv(Rθz), y(x1, x2) = v(x1, εx2).
Lemma 2.1 and Theorem 2.3 apply in particular to the functionals (3.2), and the following corollary can be proved by standard arguments of the theory of Γ-convergence. In order to state it, we define Jθ
T : A → R as Jθ T(d1, d2, d3) := ˆ I QTθ(d01· d3, d02· d3) dx1, (3.3)
where A is the class of orthonormal frames defined in (2.11) and QTθ is defined as in (2.13) with Lθ
T in place of Lθ (see also above).
Corollary 3.1. If (yε) ⊂ W2,2iso,ε(S, R
3) is a sequence of minimisers ofJθ ε,T then,
up to a subsequence, there exist y ∈ W2,2(I, R3) and a minimiser (d1|d2|d3) ∈ A of
Jθ
T with d1= y0 such that
yε* y in W2,2(S, R3), ∇εyε* (d1|d2) in W1,2(S, R3×2), and Ay,ε* d01· d3 d02· d3 d02· d3 γ
in L2(S, R2×2sym), for some γ ∈ L2(S, R3). (3.4) Moreover, min W2,2iso,ε(S,R3)J θ ε,T −→ min A J θ T.
The following proposition gives the explicit expression of QTθ. Proposition 3.2. QTθ is a continuous function given by
QTθ(α, β) = 4c1k aθα − bθβ + c1k2 2 −c1 ca 2 θ + ¯eT, in DT 4 cβ2− c 1kbθβ + c1k2 2 −c1 ca 2 θ + ¯eT, in UT c(α2+βα22)2 + 2c1k aθα 2−β2 α − 2bθβ + 2c1k2+ ¯eT, in VT,
where aθ and bθ are defined in (3.1), and
DT := n (α, β) ∈ R2: c1 c kaθα > β 2+ α2o, (3.5) UT := n (α, β) ∈ R2: c1 c kaθα ≤ β 2− α2o, (3.6) VT := R2\ (DT ∪ UT). (3.7)
Proof. It suffices to calculate the coefficients l1( ¯AθT) and l2(α, β, ¯AθT) for the case
of the twist geometry. An easy computation shows that l1( ¯AθT) = −2c1kaθ
and
l2(α, β, ¯AθT) = 2c1kaθα − 4c1kbθβ + 2c1k2+ ¯eT.
The result now follows by Proposition 2.2.
Note that when k = 0 (and c = 1), modulo the constant ¯eT, the expression for
QTθ in Proposition 3.2 reduces to expression (1.5) in [19], namely,
Q(α, β) := ( 4 β2 if α2≤ β2 (α2+β2)2 α2 if α 2> β2. (3.8)
Indeed, in this case
DT = Ø, UT =(α, β) ∈ R2: α2≤ β2 . (3.9)
For k > 0, it is natural to distinguish the case θ = π/4 (and, similarly, the case θ = 3π/4) from all the other cases. Indeed, we have aπ/4= 0 and bπ/4= 1, so that
DT and UT are again given by (3.9), and
QTπ/4(α, β) = (
4 cβ2− c
1kβ + 2c1k2+ ¯eT, if α2≤ β2
c(α2+βα22)2 − 4c1kβ + 2c1k2+ ¯eT, if α2> β2.
We refer the reader to the last picture of Figure 2 (counting clockwise from the top left). Observe that for all θ ∈ [0, π/2) \ {π/4, 3π/4}, setting ρ := c1kaθ/(2c), we
have that DT coincides with the (open) disk (α − ρ)2+ β2 < ρ2 and UT with the
(closed) region inside the hyperbola (α + ρ)2− β2= ρ2, see the other three pictures
of Figure 2.
We want to examine the minimisers of QTθ. Observe that in the case k = 0 the above function (α, β) 7→ Q(α, β) is minimised by (0, 0). This means that the 1D energy (3.3) is minimised by a trivial configuration with zero flexure and torsion, i.e. in the absence of applied loads, the rod exhibits no spontaneous flexure or torsion. When instead k > 0, we have that the minimisers of QTθ are non-trivial (in fact, they lie on a segment) and spontaneous flexure and torsion of the rod become possible. This is the content of the following lemma.
0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.2 0.1 0.0 0.1 0.2 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.2 0.1 0.0 0.1 0.2 0.3 0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.2 0.1 0.0 0.1 0.2 0.2 0.1 0.0 0.1 0.2 0.2 0.1 0.0 0.1 0.2 0.3
Figure 2. Phase diagrams in the (α, β)-plane, with level curves of QTθ. The white lines emphasize the boundary of VT (cf. (3.5)–
(3.7)) and the red lines the set of minimisers. The pictures cor-respond to the cases θ = 0, θ = π/8, θ = π/4, and θ = π/2, respectively, counting clockwise from top left.
Lemma 3.3. For every 0 ≤ θ < π, QTθ attains its minimum value precisely on the segmentαT θ,1, α T θ,2 × {βθ}, where βθ:= k c1 2 c sin 2θ, and αTθ,1:= −k c1 2 c (1 + cos 2θ), α T θ,2:= k c1 2 c(1 − cos 2θ). Moreover, min R2 QTθ = c1k2 2 −c1 c + ¯eT. (3.10)
Note that the segment αT
θ,1, αTθ,2 × {βθ} is a subset of UT connecting the two
branches of the hyperbola c1
ckaθα ≤ β
2− α2 (a degenerate hyperbola in the case
Proof. Consider the nontrivial case k 6= 0. A straightforward computation shows that the points (α, β) ∈ αT
θ,1, αTθ,2 × {βθ}, which lie in the interior of UT, are local
minimisers, and that QTθevaluated at each of these points gives the value (3.10). At
the same time, when DT 6= Ø, we have that ∇Qθ(α, β) 6= 0 for every (α, β) ∈ DT,
because aθ and bθ can never vanish simultaneously. Moreover, a point (α, β) lying
in the interior of VT is a critical point of Q θ T iff c α − cβ 4 α3 + c1k aθ β2 α2 + c1k aθ = 0, (3.11) cβ 3 α2 + c β − c1k aθ β α− c1k bθ = 0. (3.12) Now, observe that in the case β 6= 0, multiplying the second equation by β/α and adding the first yields
c (α2+ β2) + c1k aθα + bθβ = 0.
At the same time, equation (3.11) is equivalent to c (β2− α2) − c
1k aθα = 0.
Summing up the last two equations gives β = c1kbθ/(2c) and in turn α = αTθ,1
or α = αTθ,2, when bθ 6= 0. In the case bθ = 0, a similar argument gives that
(−c1k/c, 0) is the solution to (3.11)–(3.12). In any case, we have obtained that the
solutions of (3.11)–(3.12) lie in ∂UT. Hence, there are no critical points of Q θ T in the
interior of UT. Other straightforward computations show that the values of Q θ
T on
∂UT \ {(αTθ,1, βθ), (αTθ,2, βθ)} are strictly smaller than the local minimum, therefore
the local minimisers are indeed global. This concludes the proof of the lemma. Remark 3.1 (Spontaneously curved LCE twist ribbons). Using the above lemma we can now find the minimisers and the minimum of our limiting functional (3.3). Indeed, minimising the integrand pointwise, we have that
min A J θ T = ` min R2×2 QTθ = ` h c1k2 2 − c1 c + ¯eT i = µ ` π4 3 1 + 2λ 1 + λ +π 4− 4π2− 48 8 η2 0 h2 0 ,
where in the second equality we have used (3.10) and in the last one the constants c1, c, k, and ¯eT have been substituted with their expressions given in terms of
the parameters of the 3D model (cf. (2.2)–(2.4)), which are amenable to direct measurement in the laboratory synthesizing the LCE specimen. The set (α, β) ∈ αT θ,1, α T θ,2 × {βθ} of the minimisers of Q θ T is given by 3 η 0 π2(1 + λ) h 0 (−aθ− 1) , 3 η0 π2(1 + λ) h 0 (1 − aθ) × 3 η 0 π2(1 + λ) h 0 bθ . (3.13) Hence, any (d1, d2, d3) ∈ A such that the flexure d10 · d3 and the torsion d02· d3are
constant and satisfy d01·d3∈ 3 η 0 π2(1 + λ) h 0 (−aθ− 1) , 3 η0 π2(1 + λ) h 0 (1 − aθ) , d02·d3 = 3 η0 π2(1 + λ) h 0 bθ,
is a minimiser of JTθ. Indeed, given two continuous functions I 3 x17→ α(x1) and
I 3 x17→ β(x1), there always exists a C1(I)-solution to the problem
In particular, when α and β are constant, identifying (d1, d2, d3) with the rotation
matrix Q = (d1|d2|d3), it is standard to see that finding a solution of (3.14) consists
in solving Q0(s) = Q(s) 0 0 −α 0 0 −β α β 0 , (3.15)
for some fixed Q(0) ∈ SO(3), and that the solution Q(s) is actually a rotation matrix. Also, once s 7→ d1(s) is given, the mid-line curve is given by
r(s) = r(0) + ˆ s
0
d1(σ)dσ (3.16)
and it is fixed up to a translation. The last two equations allow us to reconstruct the rod configuration through its mid-line curve (3.16) and the orientation of the directors given by the solution to (3.15).
For the convenience of the reader, we consider explicitly two cases: θ = 0 : (d01· d3, d02· d3) ∈ − 6 η0 π2(1 + λ) h 0 , 0 × {0} , θ = π/4 : (d01· d3, d02· d3) ∈ − 3 η0 π2(1 + λ) h 0 , 3 η0 π2(1 + λ) h 0 × 3 η 0 π2(1 + λ) h 0 . Some examples of the corresponding rod configurations are shown in Figures 3 and 4, respectively. The configurations are rendered by plotting the mid-line curve r(s) thickened along the direction d2(s) by a fixed, small amount.
Figure 3. Some examples of minimal energy configurations for the 1D model (3.3) (where the energy density is given by Proposi-tion 3.2), for the case θ = 0. From left to right, the configuraProposi-tions are characterized by an increasing (constant) value of α = d01· d3
in the admissible interval − 6 η0
π2(1+λ) h0, 0. In particular, in the
last picture α = 0. We plot the mid-line of the curve s 7→ r(s) (red line), and, at each point r(s), a segment along the director d2(s)
(yellow).
We note that the minimisers predicted by our limiting model are in agreement with [31, Figure 3], where minimum-energy configurations of self-shaping synthetic systems made of swelling hydrogels with oriented reinforcements are shown.
Figure 4. Two examples of minimal energy configurations for the 1D model (3.3), where the energy density is given by Proposition 3.2, for θ = π/4. They are characterized by the same constant value β = d02· d3, and by two constant values of α = d01· d3, taken
in the admissible interval − 3 η0
π2(1+λ) h
0,
3 η0
π2(1+λ) h
0. In particular,
the second configuration corresponds to α = 0 and its mid-line is a straight line. In the other configuration the mid-line is a helix. We plot the mid-line of the curve s 7→ r(s) (red line), and, at each point r(s), a segment along the director d2(s) (yellow).
4. The splay-bend case
In the splay-bend case, the θ-dependent target curvature tensor ¯Aθ defined in
(2.7) is
¯ AθS = k
− cos2θ sin θ cos θ
sin θ cos θ − sin2θ
and det ¯Aθ
S = det ¯AS = 0. The functional defined in (2.8) becomes
Eθ ε,S(v) = 1 ε ˆ Sε n c|Av(z)|2+ LθS(Av(z)) o dz, with LθS(A) := −2c1A · ¯ASθ + 2c2k trA + ck2+ ¯eS.
Also, the rescaled energy (2.10) is now given by Jθ
ε,S(y) :=
ˆ
S
n
c|Ay,ε(x)|2+ LθS(Ay,ε(x))
o dx, for every y ∈ Wiso,ε2,2 (S, R3), andJSθ: A → R is defined as
Jθ
S(d1, d2, d3) :=
ˆ
I
QSθ(d01· d3, d02· d3) dx1, (4.1)
where QSθ is given by (2.13), with Lθ
S in place of L
θ. The counterpart of
Corol-lary 3.1 holds for the splay-bend case: it is sufficient to replace Jε,Tθ and JTθ by Jθ
ε,S andJSθ, respectively, in the statement of Corollary 3.1.
Note that thanks to Lemma 4.2 below, the minimum of Jθ
ε,S in W
2,2
iso,ε(S, R
3)
and the minimum ofJθ
S in A can be computed explicitly, so that the convergence
of the former to the latter can be checked by hand. The 2D minimizing rescaled second fundamental forms can be computed as well, together with the 1D minimal flexure and torsion, and with γ (cf. (3.4)). Therefore, also the convergence of the rescaled second fundamental forms can be checked by hand.
To proceed with our analysis, we note that ¯Aθ
S can be alternatively written as
¯ AθS =1 2 ¯ AθT − I and in turn LθS(A) = −c1A · ¯ATθ + k(c + c2) trA + ck2+ ¯eS.
Hence, setting dθ= c1aθ− c − c2, we have that
QSθ(α, β) = min
γ∈Rf (γ),
where
f (γ) := c(α2+ 2β2+ γ2) + 2c|αγ − β2| − kdθ(α + γ) + 2kc1(aθα − bθβ) + ck2+ ¯eS,
(4.2) recalling that aθ:= cos 2θ and bθ:= sin 2θ.
Proposition 4.1. QSθ is a continuous function given by
QSθ(α, β) = 2c1k aθα − bθβ + ck2 1 − d2θ 4c2 + ¯eS, in DS 4cβ2− 2kdθα + 2c1k aθα − bθβ + ck2 1 − d2θ 4c2 + ¯eS, in US c(α2+βα22)2 − kdθα 2+β2 α + 2c1k aθα − bθβ + ck 2+ ¯e S, in VS, where dθ:= c1aθ− c − c2, and DS := (α, β) ∈ R2: k 2cdθα > β 2+ α2 , (4.3) US := (α, β) ∈ R2: k 2cdθα ≤ β 2− α2 , (4.4) VS := R2\ (DS∪ US). (4.5)
Proof. By (4.2), we immediately deduce that l1( ¯AθS) = −kdθ
and
l2(α, β, ¯AθS) = −kdθα + 2kc1(aθα − bθβ) + ck2+ ¯eS.
The result now follows by substituting these values in Proposition 2.2. We refer the reader to Figure 5 where plots of the level curves of QSθare displayed for a selection of angles θ.
Remark 4.1. Setting QS,1θ := 2c1k aθα − bθβ + ck2 1 − d 2 θ 4c2 + ¯eS,
we have that QSθ= QS,1θ in DS, that
QSθ = QS,2θ (α, β) := 4cβ2− 2kdθα + Q θ S,1(α, β) in US, and that QSθ = cα 2+ β2 α − kdθ 2c 2 +QS,1θ (α, β) = cβ 2− α2 α − kdθ 2c 2 +QS,2θ (α, β) in VS.
Since the squares in these expressions vanish on ∂DS and ∂US, respectively, this
1.5 1.0 0.5 0.0 1.0 0.5 0.0 0.5 1.0 1.5 1.0 0.5 0.0 1.0 0.5 0.0 0.5 1.0 1.5 1.0 0.5 0.0 0.5 0.0 0.5 1.5 1.0 0.5 0.0 1.0 0.5 0.0 0.5 1.0
Figure 5. Phase diagrams in the (α, β)-plane, with level curves of QSθ. The white lines emphasize the boundary of VT (cf. (4.3)–(4.5))
and the red lines the set of minimisers. The pictures correspond to the cases θ = 0, θ = π/8, θ = π/4, and θ = π/2, respectively, counting clockwise from top left.
We now focus on the minimisers of QSθ. As for the twist case, when k = 0 and up to additive and multiplicative constants, the function QSθ reduces to the function defined in (3.8), which is minimised at (0, 0). When instead k > 0, differently from the twist case, we have that for every θ the minimiser of QSθ is a (θ-dependent) single point, in view of the following lemma.
Lemma 4.2. For every 0 ≤ θ < π, QSθ is minimised precisely at (αS
θ, βθS), where αSθ := −k 2 1 + cos 2θ, β S θ := k 2sin 2θ. Moreover, min R2 QSθ = ¯eS. (4.6)
Proof. Consider the nontrivial case k 6= 0. A straightforward computation shows that there are no critical points of QSθ in the interior of DS or US, for any value of
the two expressions for QSθ in VS given in Remark 4.1 and setting ∇Q θ S = 0 yields 2cα 2+ β2 α − kdθ 2c β2− α2 α2 = 2c1kaθ (4.7) 2cα 2+ β2 α − kdθ 2c β α = c1kbθ. (4.8)
We first examine the cases θ = 0 and θ = π/2. In these cases, bθ= 0 and |aθ| = 1,
so that from (4.8) we get β = 0, since (α2+ β2)/α 6= kdθ/(2c) in the interior of VS.
Setting β = 0 in (4.7) we then get α = −2ck(c1aθ+ c + c2). Now, an easy calculation
shows that in the case θ = 0 the point (αS
0, β0S) lies in the interior of VS and it is
the global minimiser of QSθ. On the other hand, in the case θ = π/2 another easy computation shows that the the point (−kc2/c, 0) lies in DS and it is thus not a
critical point of our function in the interior of VS. Then, comparing the values of
QSπ/2 on the boundaries of DS and US, we find that the minimum is achieved at
(αS
π/2, β
S
π/2) = (0, 0). Moreover, it can be readily checked that (4.6) holds for θ = 0
and θ = π/2.
Consider now an arbitrary θ ∈ (0, π) \ {π/2} and note that in this case bθ6= 0
and |aθ| < 1. As before, we search for critical points of Q θ
S in the interior of VS.
From (4.8) we get in particular β 6= 0. Therefore, we may divide (4.7) by (4.8) getting β2− α2 α = 2 aθ bθ β, (4.9)
and in turn that |α| = |aθα − bθβ|. Hence, either α = aθα − bθβ or α = bθβ − aθα.
Suppose that the former case holds true or, equivalently, that β
α= aθ− 1
bθ
. (4.10)
Before proceeding, consider the second expression for QSθ in VS given in Remark
4.1, namely QSθ = c β2− α2 α − kdθ 2c + QS,2θ (α, β).
Differentiating it with respect to β and setting ∂βQ θ S= 0 yields 2cβ 2− α2 α − kdθ 2c β α= c1kbθ− 4cβ. This expression, coupled with (4.9) and (4.10), easily gives
(α, β) =k 2 b2 aθ− 1 , bθ .
Recalling the definition of aθ and bθ, this point coincides with (αSθ, βSθ) defined in
the statement, and lies in the interior of VS. Supposing now that α = bθβ − aθα
and proceeding in a similar manner returns the point −kc2 2c b2 θ aθ+ 1 , bθ
which belongs to DS. Hence, the only critical point in the interior of VS is (αSθ, β S
θ).
Other computations show that this is indeed the global minimiser of QSθ and that
Remark 4.2 (Spontaneously curved LCE splay-bend ribbons). In view of the above lemma, we have that the minimum of our limiting functional (4.1) is
min A J θ S = ` min R2×2 QSθ = ` ¯eS = µ ` (1 + λ) π4− 12 32 η2 0 h2 0 ,
where in the second equality we have used (4.6) and in the last one the constant ¯eS
has been replaced by its expression given in terms of the 3D parameters (the first expression in (2.3)). Also, the minimisers of Jθ
T are all the triples (d1, d2, d3) ∈ A
such that (d01· d3, d02· d3) = k 2 − 1 − cos 2θ, sin 2θ = 3 η0 π2h 0 − 1 − cos 2θ, sin 2θ, θ ∈ [0, π),
where in the second expression we have used (2.2). For the convenience of the reader, we list some cases explicitly:
θ = 0 : (d01· d3, d02· d3) = 3 η0 π2h 0 (−2, 0); θ = π/8 : (d01· d3, d02· d3) = 3 η0 π2h 0 − √ 2 + 2 2 , √ 2 2 ! ; θ = π/4 : (d01· d3, d02· d3) = 3 η0 π2h 0 (−1, 1); θ = π/2 : (d01· d3, d02· d3) = (0, 0).
The (minimal energy) configurations of the corresponding rods can be computed and plotted as described in Remark 3.1. Except for the case θ = π/2, they are all spiral ribbons, with flexure and torsion given by explicit formulas.
Remark 4.3. Generically, by Taylor-expanding at order two around a minimum the energy densities we have derived, one obtains a 1D free-energy functional of the form
J = ˆ −`/2
−`/2
C1(α − α0)2+ C2(β − β0)2+ C3αβ + C4, (4.11)
where α and β stand here for the flexure and torsion functions d01· d3 and d02· d3,
respectively. For example, for θ = 0 in the splay-bend case, we have α0 = −k,
β0= 0, C1= c, C2= 2c1, C3= 0, and C4= ¯eS. By contrast in some cases (e.g. for
θ = π/2), the Hessian at the minimiser is not positive definite and the quadratic approximation (4.11) fails.
The 1D energies obtained in Section 2 and, in particular, those obtained for twist and splay-bend LCE ribbons in Sections 3 and 4 are, however, far from being quadratic. The difference between such energies and (4.11) is, in most cases, quite dramatic and it becomes apparent far away from the minima. This may lead to the fact that interesting behaviour in the presence of externally applied loads is missed if, instead of using the 1D functionals we have derived, one replaces them with those obtained by a quadratic approximations of the integrands, such as in (4.11). Acknowledgements. We gratefully acknowledge the support by the European Research Council through the ERC Advanced Grant 340685-MicroMotility. This work was started after an inspiring lecture given by Prof. R. Paroni at “Physics and Mathematics of Materials: current insights”, an international conference in honour of the 75th birthday of Paolo Podio-Guidugli held at Gran Sasso Science Istitute (L’Aquila) in January 2016.
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